The Uncertainty Principle

1 THE UNCERTAINTY PRINCIPLE (1927) Werner Heisenberg - Nobel Prize in Physics (1932) at age of 31! Mentor: Niels Bohr The values of par�cular pairs of observables cannot be determined simultaneously with arbitrarily high precision in quantum mechanics. Examples of pairs of observables that are restricted in this way are momentum and posi�on, and energy and �me; such pairs are referred to as complementary. The quan�ta�ve expressions of the Heisenberg uncertainty principle can be derived by combining the de Broglie rela�on p = h / λ and the Einstein rela�on E = hv with proper�es of all waves. The de Broglie wave for a par�cle is made up of a superposi�on of an infinitely large number of waves of the form ( ) ( ) , sin 2 sin 2 x x t A vt A x vt ψ π λ π κ = = where A is amplitude and κ is the reciprocal wavelength. We consider one spa�al dimension for simplicity. The waves that are added together have infinitesimally different wavelengths. This superposi�on of waves produces a wave packet shown to the right. 1 1 4 x x κ λ π =∆ (1) (a) weakly localized 1 4 t v π ∆ ∆ (2) (b) strongly localized where Δ x is the extent of the wave packet in space, Δ κ is the range in reciprocal wavelength, Δ v is the range in frequency, and Δ t is a measure of the �me required for the packet to pass a given point. The Δ's in these equa�ons are actually standard devia�ons.
2 If at a given �me the wave packet extends over a short range of x values, there is a limit to the accuracy with which we can measure the wavelength. If a wave packet is of short dura�on, there is a limit to the accuracy with which we can measure the frequency. S ubs�tu�ng the de Broglie rela�on in equa�on (1) , 1/λ = p x / h for mo�on in the x direc�on, then 1 4 x p x h π 2 x x p ħ = h /2 π ( h bar) Another form of the uncertainty principle may be derived by subs�tu�ng E = hv in equa�on (2) and it yields 1 4 E t h π ∆ ∆ 2 t E ∆ ∆
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